As I see it, the definition of kinetic energy $$T= {1\over2} m u^2 \text { where $u<<c$}$$ comes by using the definition of work $$W= {\int F\cdot\ dx }$$ and we use for the meaning of F(force) the Second Law of Newton: $$F={dp\over dt}=ma$$
Do I understand correctly that the kinetic energy from this point and on becomes a connection link between Newtonian mechanics, theoretical mechanics(Lagrange, Hamilton) and relativity?
If so, this the question: Can there be a definition of energy without the law of Newton?(it seems to me that to use the law of newton is not wrong but strange to the point of view that comes with theoretical mechanics. We remain somehow bounded to a definition of kinetic energy associated with a law that makes the mass something a little more fundamental from let's say charge-by fundamental I mean that something without mass, Newton says it cannot move or have any type of interaction. But if the charge is another way of interaction-via the electromagnetic fields or potentials- shouldn't we search for the possibility of defining the kinetic energy via the charge. What should become of the potentials then? And why mass in the Newton's law?).
Thank you.
PS:if anyone has a recommendation for further study, he can suggest it.